Methods of homological algebra同调代数方法
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作者： Sergei I. Gelfand，Yuri I. Manin著
出 版 社： 湖南文艺出版社
出版时间： 2003-1-1字数：版次：页数： 372印刷时间： 2003/01/01开本： 16开印次：纸张： 胶版纸I S B N ： 9783540435839包装： 精装内容简介
Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections.
目录
I. Simplicial Sets
1.1 Triangulated Spaces
1.2 Simplicial Sets
1.3 Simplicial Topological Spaces and the Eilenberg-Zilber Theorem
1.4 Homology and Cohomology
1.5 Sheaves
1.6 The Exact Sequence
1.7 Complexes
II. Main Notions of the Category Theory
II.1 The Language of Categories and Functors
II.2 Categories and Structures, Equivalence of Categories
II.3 Structures and Categories. Representable Functors
II.4 Category Approach to the Construction of Geometrical Objects
II.5 Additive and Abelian Categories
II.6 Functors in Abelian Categories
III. Derived Categories and Derived Functors
III.1 Complexes as Generalized Objects
III.2 Derived Categories and Localization
III.3 Triangles as Generalized Exact Triples
III.4 Derived Category as the Localization of Homotopic Category
III.5 The Structure of the Derived Category
III.6 Derived Functors
III.7 Derived Functor of the Composition. Spectral Sequence
III.8 Sheaf Cohomology
IV. Triangulated Categories
IV.1 Triangulated Categories
IV.2 Derived Categories Are Triangulated
IV.3 An Example: The Triangulated Category of A-Modules
IV.4 Cores
V. Introduction to Homotopic Algebra
V.1 Closed Model Categories
V.2 Homotopic Characterization of Weak Equivalences
V.3 DG-Algebras as a Closed Model Category
V.4 Minimal Algebras
V.5 Equivalence of Homotopy Categories
References
Index